Beskjeder

I will give a special problemsession on Friday June 2. I will do problem 3 and 4 of exam MA 252, 17/12- 2001, Problem 1 and 2 of exam MA 252 10/12-1999 and Problem 3 of exam MA 252 8/12- 2000. Links to the text of these exams are

here

Final examination will take place during Tuesday June 6. and (if necessary) Wednesday June 7.

28/4 I finished section 5.5 and Theorem 6.1.3, Then I talked about cochain maps and cochain homotopies, and I started on section 6.3 and proved Lemma 6.3.1. 5/5 I finished section 6.3 and I also lectured 6.4. Last Friday (12/5) I calculated the de Rham cohomology of the spheres and tori (6,5) and finished with lecturing 6.7. This will end the syllabus. (You will find an updated revised version of the syllabus on the main homepage ) Next Friday (19/5) I will do problem 6.4, 6.5 and start on the exam MA 252 4/12 2002. You will find the text for this exam

here

Last friday (21/4) I started the lecture showing that the version of Stokes Theorem that you (probably) know from the vector calculus (where M is a smooth surface with boundary in three space) can be derived from the general version of Stokes Theorem given in this course. I then gave some applications of Stokes Theorem (5.4.2, 5.4.3 and 5.4.4 in Barden and Thomas). Then I started on 6.1 and defined de Rham cohomology for a manifold, and I proved Prop 6.1.2., formulated Theorem 6.1.3 and then jumped back to section 5.5 which I need for the proof of 6.1.3. I proved 5.5.1 and I will start on 5.5.2 today,and then finish section 5.5, 6.2 and (hopefully) 6.3.

17/3, 24/3 and 31/3 I covered the material in 4.1 (I think that 4.1 contains few details soo I lectured the multilinear algebra from Spivak instead), 4.2, 4.3 (here (as examples) I proved that the spheres were orientable and that projective spaces are orientable if and only if the dimension is odd) and 5.1 and 5.2. Today I talked about manifolds with boundaries (5.3), integration of forms on compact orientable manifolds (4.4). Then I formulated and (tried to) proved Stokes theorem (5.4) (in the end of the proof I unfortunately ran out of time so the calculations became rather sketchy, but since the proof in the textbook is rather readble I will not repeat the proof.) There will (of course) be no lecture next friday, but 21/4 I will do some applications of Stokes Theorem and I will then defer 5.5 and start on chapter 6

17/2 and 24/2 I defined the tangent and cotangentbundle and I talked about vectorbundles in general (3.2). 3/3 I defined orientability for a vectorbundle in general, and I also talked about vectorfields and 1-forms (2.5). During these weeks I also finished the remaining exercises I gave 10/2: 10/3 I lectured 2.6 and 2.7. 17/3 I will start on chapter 4. The third hour I will look at exercise 2.3.

3/2: I proved the Theorem 1.3.10 (local embedding theorem). Note that this theorem is (obviously) wrongly stated in the text book.A correct formulation of the theorem can you find

here

I lectured the remaining part of 1.3, then i started on section 1.4,constructed bump functions, proved Theorem 1.4.3 (that every compact manifold can be embedded into some Euclidean space. Then I refered Whitney Embedding Theorem (that every smooth m-manifold can be immersed into R^2m and embedded into R^(2m+1)).

10/2: I constructed partions of unity on a smooth manifold and showed that manifols are paracompact (section 1.5). I started on chapter 2 lectured 2.1 and 2.2. On 17/2 I will start lecturing 2.3, then I will jump to 3.2 and talk about vectorbundles in general. In the third hour I will do some exercises namely 1.3, 1.4 and 1.6 together with the following

two

20/1: I defined smooth manifolds, gave examples of such manifolds and defined and gave examples of smooth maps between smooth manifolds. The lecture covered more or less the material in the textbook up to 1.3.

27/1: I reviewed the Inverse Fuction Theorem, proved the Submersion and Implicit Function Theorem (1.3.1, 1.3.2 and 1.3.3). I defined a smooth submanifold of smooth manifold and showed that the inverse image of a regular value (for some smooth map) is a smooth submanifold. I gave examples of smooth submanifolds of Euclidean spaces of this kind. The lecture covered the material of section 1.3 in the textbook up to 1.3.10 (which will be the starting point of the next lecture).

Velkommem til MAT 4520. F?rste forelesning blir fredag 20. januar. Som det vil fremg? av pensumlisten bruker vi samme l?rebok som ifjor(Barden og Thomas) og pensumet er i utgangspunktet det samme. I skrivende stund er det tilsammen 8 eksemplarer av boka i bokhandelen. Skulle boka bli utsolgt f?r undervisningen begynner s? send meg en mail s? raskt som mulig s? jeg kan ta det opp med bokhandelen. H.B. (Hans Brodersen)